Online Function Tracking

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Online Function Tracking with Generalized
Penalties
Marcin Bienkowski Institute of Computer Science,
University of Wroclaw, Poland
Stefan Schmid Deutsche Telekom Laboratories / TU
Berlin Germany
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Lets start with an example
round 3
round 4
round 1
round 2
Knows the measurement with absolute error
11-10 1
Knows the measurement with absolute error
20-25 5
measures 5
measures 10
measures 11
measures 20
5
10
25
Sends update (5)
Sends update (10)
Sends update (25)
Sensor node decides not to send anything
A cost is associated both with sending updates
and inaccuracies
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The model and notation

• Algorithms state the value stored at base
  station
• In one round
• New integer value A is observed at sensor node.
• Sensor may send an update to base station
  change state for cost C.
• We pay penalty ? (A state). ? is called
  penalty function.

A
state
state
update
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Note from one reviewer

• I dont understand the story behind this model
• but the problem in this paper is to approximate
  an arbitrary gradually revealed function by a
  piecewise constant function

• 1-lookahead model
• observe
• change state
• pay for inaccuracies

time
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The goal

• Input sequence ? of observed/measured integral
  values
• Output schedule of updates
• Cost the sum of update and penalty costs.
• Goal Design effective online algorithms for an
  arbitrary input.
• Online we do not know the future measurements.
• Effective ALG is R-competitive if for all ?,
• OPT optimal offline algorithm
• R competitive ratio, subject to
  minimization.

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Our contributions

• Competitive algorithms for various penalty
  functions.
• When ? is concave
• deterministic -competitive
  algorithm.
• When ? is convex
• deterministic -competitive
  algorithm.
• Matching lower bounds for randomized algorithms.

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Related results

• Variant of the data aggregation.
• If observed values change monotonically, the
  problem is a variant
• of the TCP acknowledgement problem
• ) O(1)-competitive solutions exist.
• Yi and Zhang SODA 2009
  -competitive algorithm for
• (their approach works also for
  multidimensional case)

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This talk

• When ? is concave
• deterministic -competitive
  algorithm.
• When ? is convex
• deterministic -competitive
  algorithm.

This talk only for the case ?(x) x
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A simplifying assumption

• Whenever algorithm in state V observes value A
• ?(V-A) C

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Algorithm MED for concave penalties

• One phase of MED
• Algorithms state vMED
• Wait till the accumulated penalties (in the
  current phase)
• would exceed C and then
• change state to vMED the median of values
  observed in this phase.

,,Accumulate-and-update paradigm
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Analysis of MED for ?(x) x (3)

• Lemma A In a phase, MED pays O(C).
• Proof
• Penalties for all rounds but the last one C.
• Sending update in the last round C
• Penalty for the last round vMED ALAST

median (transmitted value)
Penalty for the last round 2C
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Analysis of MED for ?(x) x (2)

• Lemma B Fix phase P. Assume that OPT does not
  update in P. Then,
• 
  , where ? COPT(P)
• Proof

n 6 observed values
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Analysis of MED for ?(x) x (3)

• Theorem MED is O(log C)-competitive
• Proof Fix an epoch E of any ?(log C) consecutive
  phases.
• By Lemma A, CMED(E) O(C log C).
• To show there exists phase P, s.t., COPT(P)
  C/4.
  
• Assume the contrary ) OPT remains at one
  state in E.
• Let X vMED vOPT.

More carefull analysis shows competitive ratio of
O(log C / log log C)
X decreases by 2/3 in each phase (by Lemma B)
(technical lemma)
OPT and MED in the same state ) OPT pays at least
C.
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Algorithm SET for convex penalties (1)

• Same accumulate-and-update paradigm as for MED,
  i.e., in one phase it
• remains at one state, vSET, till the accumulated
  penalties would exceed C
• and then changes state to vSET.
• For any phase, SET pays O(C)
  (same argument as for MED)
• How to choose vSET to build a lower bound for
  OPT?
• If OPT changes state it pays C
• We have to assure that staying in a fixed state
  is expensive

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Algorithm SET for convex penalties (2)
At the beginning, SET changes state to the first
observed value. S vSET - ?, vSET ? set
of good states

• A(x) the sum of penalties in
• the current phase if state x.
• At the end of the phase
• states x A(x) gt C are
• removed from S.
• vSET center of new S

A is convex
A
At least half of S is removed!
In O(log ?) phases OPT pays at least C
C
vSET
vSET-?
vSET?
new S
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Final remarks

• When ? is concave deterministic
  -competitive algorithm
• When ? is convex deterministic
  -competitive algorithm.
• Matching lower bounds for randomized algorithms.

• Presented case ?(x) x, but only triangle
  inequality of ? was used.
• Same algorithm works for
• Concave ?
• ? where triangle inequality holds approximately,
  e.g., ?(x) xd

Possible to add hard constraints the
difference between measured value and state must
not be larger than T.
Randomization does not help (even against
oblivious adversaries)
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Thank you for your attention!